What is Control Theory?

What is Control Theory?What is Feedback? How does it work? What is Stability?

John D. Fox, Claudio Rivetta

Stanford Linear Accelerator CenterWork supported by the DOE under contract # DE-AC02-76SF00515

June 2012

John D. Fox, Claudio Rivetta (SLAC) What is Control Theory? June 2012 1 / 27

Outline

1 Feedback

2 Signals in the Time and Frequency domainsFourier Transforms

3 Linear Time Invariant SystemsImpulse Response, ConvolutionA Quiz


Feedback

Origins


Feedback

Feedback BasicsACAS Workshop and School Dec. 2010

Feedback basics

The objective is to make the outputof a dynamic system (plant) behave ina desired way by manipulating inputor inputs of the plant.

Regulator problem - keep small orconstant

Servomechanism problem - makefollow a reference signal

Feedback controller acts to reject the external disturbances.

The error between and the desired value is the measure of feedback system performance. There aremany ways to define the numerical performance metric

• RMS or maximum errors in steady-state operation

• Step response performance such as rise time, settling time, overshoot.

An additional measure of feedback performance is the average or peak actuator effort. Peak actuatoreffort is almost always important due to the finite actuator range.

Feedback system robustness - how does the performance change if the plant parameters or dynamicschange? How do the changes in sensors and actuators affect the system?

controller

sensors

actuators Plantr u y

external disturbancesy

y

yr

y


Feedback

Feedback Basics - engine control via Watt’s regulator

SetpointControllerActuatorPlant

DisturbanceSensor


Feedback

Feedback and the Motor City


Feedback

Oxygen sensor, feedback for mixture control


Feedback

Stability


Feedback

The simplest controller

How do we pick the gain?


Feedback

The simplest controller, gets complicated


Feedback

Can we do better?

What should this do to the frequency response? What should this doto the time ( Step) response?


Feedback

we can do better, but how much?

What should this do to the frequency response? What should this doto the time ( Step) response?John D. Fox, Claudio Rivetta (SLAC) What is Control Theory? June 2012 14 / 27

Feedback

The PI controller

We want to learn how to designcontrollers, and understandstability. We want to understandhow a controller changes thedynamics of a system.


Feedback

Feedback, basically, it can get very complicatedACAS Workshop and School Dec. 2010

Feedback basics

The objective is to make the outputof a dynamic system (plant) behave ina desired way by manipulating inputor inputs of the plant.

Regulator problem - keep small orconstant

Servomechanism problem - makefollow a reference signal

Feedback controller acts to reject the external disturbances.

The error between and the desired value is the measure of feedback system performance. There aremany ways to define the numerical performance metric

• RMS or maximum errors in steady-state operation

• Step response performance such as rise time, settling time, overshoot.

An additional measure of feedback performance is the average or peak actuator effort. Peak actuatoreffort is almost always important due to the finite actuator range.

Feedback system robustness - how does the performance change if the plant parameters or dynamicschange? How do the changes in sensors and actuators affect the system?

controller

sensors

actuators Plantr u y

external disturbancesy

y

yr

y


Signals in the Time and Frequency domains Fourier Transforms

Time and Frequency Domains

Fourier transformsA function f(x) may be Fourier transformed into a function F(s),

F (s) =∫ ∞−∞

f (x)e−i2πxsdx (1)

and likewise a function F(s) can be transformed into a function f(x)

f (x) =∫ ∞−∞

F (s)ei2πxsds (2)

The Laplace transform is related to the Fourier Transform but involvesan integral from 0 to infinity




Discrete Fourier TransformFor systems involving discrete samples of data, such as from samplingcircuits or from samples taken from circulating bunches, thediscrete-time Fourier transform is similar

F (ν) =1N

N−1∑τ=0

f (τ)e−i2π(ν/N)τ (3)

f (τ) =N−1∑ν=0

F (ν)ei2π(ν/N)τ (4)

There is a related transform, the Z transform, which is the discrete-timeequivalent of the Laplace transform




Convolution of two functionsThe convolution of two functions f(x) and g(x) is defined as f (x) ? g(x)

f (x) ? g(x) =∫ ∞−∞

f (u)g(x − u)du (5)

In pictorial form



Common Transform Pairs ( from Bracewell)


Linear Time Invariant Systems


If a system converts an input u(t) into an output y(t)

y(t) = L [u(t)] (6)

the system is linear if for two constants a1 and a2

L [a1u1 + a2u2] = a1L [u1(t)] + a2L [u2(t)] . (7)

The response of two inputs is the superposition of the individualoutputs. If an input is only a single frequency ω, the output can onlycontain that single frequency ω.




A system is time invariant if for a time delay δ the output has shiftinvariance, or that

L [u(t)] = y(t) (8)

L [u(t − δ)] = y(t − δ) (9)


Linear Time Invariant Systems Impulse Response, Convolution

Impulse response of LTI system

The impulse response I(t) of a system is found by exciting the systemwith a δ-function in the time domain.

LTI

I(t)(t)

5-2000 8545A1

for a general input u(t) the output is a convolution

y(t) = u(t) ? I(t) (10)



Frequency Response of LTI system

Frequency response H(s) is the transfer function in the frequencydomain. Measured by network analyzer via magnitude and phase vs.frequency.

LTI

H(s)A(s)

5-2000 8545A2

For a general input in the frequency domain I(s) the output O(s) is theproduct

O(s) = H(s)I(s) (11)



Frequency Response and Time Response relationship

The time response is also the inverse transform of the product of theFourier transform of the input u(t) and the frequency response H(s)

y(t) = u(t) ∗ I(t) (12)

y(t) = IFT [FT (u(t))H(s)] (13)

For an LTI system, we can measure in either domain, and compute theresponse via appropriate convolutions, transforms or inversetransforms


Linear Time Invariant Systems A Quiz

A Quiz on LTI Systems

Consider this simple circuit - a voltage divider

R

R

VoutVin

5-2000 8545A16

Is this an LTI system? Is it ALWAYS an LTI system?




Consider this simple circuit - a high-pass filter

C

R

VoutVin

5-2000 8545A17

Here the output is frequency dependent. Is this an LTI system?




Consider this simple circuit - a diode clipper ( a limiter)

R

VoutVin

5-2000 8545A18

Is this an LTI system? When? What output frequencies are present foran input at ω?. Two signals ω1 and ω2? Does it have an ImpulseResponse I(t) ?


What is Control Theory?

Documents

Transcript of What is Control Theory?